A059297 - OEIS

%I #97 Sep 08 2022 08:45:02

%S 1,0,1,0,2,1,0,3,6,1,0,4,24,12,1,0,5,80,90,20,1,0,6,240,540,240,30,1,

%T 0,7,672,2835,2240,525,42,1,0,8,1792,13608,17920,7000,1008,56,1,0,9,

%U 4608,61236,129024,78750,18144,1764,72,1,0,10,11520,262440

%N Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 1.

%C T(n,k) = C(n,k)*k^(n-k) is the number of functions f from domain [n] to codomain [n+1] such that f(x)=n+1 for exactly k elements x of [n] and f(f(x))=n+1 for the remaining n-k elements x of [n]. Subsequently, row sums of T(n,k) provide the number of functions f:[n]->[n+1] such that either f(x)=n+1 or f(f(x))=n+1 for every x in [n]. We note that there are C(n,k) ways to choose the k elements mapped to n+1 and there are k^(n-k) ways to map n-k elements to a set of k elements. - _Dennis P. Walsh_, Sep 05 2012

%C Conjecture: the matrix inverse is A137452. - _R. J. Mathar_, Mar 12 2013

%C The above conjecture is correct. This triangle is the exponential Riordan array [1, x*exp(x)]. Thus the  inverse array is the exponential Riordan array [ 1, W(x)], which equals A137452. - _Peter Bala_, Apr 08 2013

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].

%H Alois P. Heinz, <a href="/A059297/b059297.txt">Rows n = 0..140, flattened</a>

%H Peter Bala, <a href="/A112007/a112007_Bala.txt">Diagonals of triangles with generating function exp(t*F(x)).</a>

%H G. Duchamp, K. A. Penson, A. I. Solomon, A. Horzela and P. Blasiak, <a href="http://arXiv.org/abs/quant-ph/0401126">One-parameter groups and combinatorial physics</a>, arXiv:quant-ph/0401126, 2004.

%H Emanuele Munarini, <a href="https://doi.org/10.2298/AADM180226017M">Combinatorial identities involving the central coefficients of a Sheffer matrix</a>, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.

%H Bruce E. Sagan, <a href="http://deepblue.lib.umich.edu/bitstream/2027.42/25401/1/0000850.pdf">A note on Abel polynomials and rooted labeled forests</a>. Discrete Mathematics 44(3): 293-298 (1983).

%H J. Taylor, <a href="https://digital.lib.washington.edu/researchworks/handle/1773/36757">Formal group laws and hypergraph colorings</a>, doctoral thesis, Univ. of Wash., 2016, p. 96, [_Tom Copeland_, Dec 20 2018].

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AbelPolynomial.html">Abel Polynomial</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IdempotentNumber.html">Idempotent Number</a>

%H Jian Zhou, <a href="https://arxiv.org/abs/2108.10514">On Some Mathematics Related to the Interpolating Statistics</a>, arXiv:2108.10514 [math-ph], 2021.

%F E.g.f.: exp(x*y*exp(y)). - _Vladeta Jovovic_, Nov 18 2003

%F Up to signs, this is the triangle of connection constants expressing the monomials x^n as a linear combination of the Abel polynomials A(k,x) := x*(x+k)^(k-1), 0 <= k <= n. O.g.f. for the k-th column: A(-k,1/x) = x^k/(1-k*x)^(k+1). Cf. A061356. Examples are given below. - _Peter Bala_, Oct 09 2011

%F The o.g.f.'s for the diagonals of this triangle are the rational functions occurring in the expansion of the compositional inverse (with respect to x) (x-t*x*exp(x))^-1 = x/(1-t) + 2*t/(1-t)^3*x^2/2! + (3*t+9*t^2)/(1-t)^5*x^3/3! + (4*t+52*t^2+64*t^3)/(1-t)^7*x^4/4! + .... For example, the o.g.f. for second subdiagonal is (3*t+9*t^2)/(1-t)^5 = 3*t + 24*t^2 + 90*t^3 + 240*t^4 + .... See the Bala link. The coefficients of the numerator polynomials are listed in A202017. - _Peter Bala_, Dec 08 2011

%F Recurrence equation: T(n+1,k+1) = Sum_{j=0..n-k} (j+1)*binomial(n,j)*T(n-j,k). - _Peter Bala_, Jan 13 2015

%F The Bell transform of [1,2,3,...]. See A264428 for the Bell transform. - _Peter Luschny_, Dec 20 2015

%e Triangle begins:

%e 1;

%e 0,  1;

%e 0,  2,   1;

%e 0,  3,   6,    1;

%e 0,  4,  24,   12,    1;

%e 0,  5,  80,   90,   20,   1;

%e 0,  6, 240,  540,  240,  30,  1;

%e 0,  7, 672, 2835, 2240, 525, 42,  1;

%e Row 4. Expansion of x^4 in terms of Abel polynomials:

%e x^4 = -4*x+24*x*(x+2)-12*x*(x+3)^2+x*(x+4)^3.

%e O.g.f. for column 2: A(-2,1/x) = x^2/(1-2*x)^3 = x^2+6*x^3+24*x^4+80*x^5+....

%p T:= (n, k)-> binomial(n, k) *k^(n-k):

%p seq(seq(T(n, k), k=0..n), n=0..12);  # _Alois P. Heinz_, Sep 05 2012

%t nn=10;f[list_]:=Select[list,#>0&];Prepend[Map[Prepend[#,0]&,Rest[Map[f,Range[0,nn]!CoefficientList[Series[Exp[y x Exp[x]],{x,0,nn}],{x,y}]]]],{1}]//Grid  (* _Geoffrey Critzer_, Feb 09 2013 *)

%t t[n_, k_] := Binomial[n, k]*k^(n - k); Prepend[Flatten@Table[t[n, k], {n, 10}, {k, 0, n}], 1] (* _Arkadiusz Wesolowski_, Mar 23 2013 *)

%o (Magma) /* As triangle */ [[Binomial(n,k)*k^(n-k): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Aug 22 2015

%o (Sage) # uses[bell_transform from A264428]

%o def A059297_row(n):

%o     nat = [k for k in (1..n)]

%o     return bell_transform(n, nat)

%o [A059297_row(n)  for n in range(8)] # _Peter Luschny_, Dec 20 2015

%Y There are 4 versions: A059297, A059298, A059299, A059300.

%Y Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc.

%Y Row sums are A000248.

%Y Cf. A061356, A202017, A137452 (inverse array), A264428.

%K nonn,easy,tabl

%O 0,5

%A _N. J. A. Sloane_, Jan 25 2001